Mathematics – Quantum Algebra
Scientific paper
2012-02-21
Mathematics
Quantum Algebra
43 pages
Scientific paper
Let $F$ denote a field, and fix a nonzero $q \in F$ such that $q^4\not=1$. The universal Askey-Wilson algebra $\Delta_q$ is the associative $F$-algebra defined by generators and relations in the following way. The generators are $A,B,C$. The relations assert that each of $A + \frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B + \frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C + \frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $\Delta_q$. The universal DAHA $\hat H_q$ of type $(C_1^\vee, C_1)$ is the associative $F$-algebra defined by generators $\lbrace t^{\pm 1}_i \rbrace_{i=0}^3$ and relations (i) $t_i t^{-1}_i=t^{-1}_i t_i = 1$; (ii) $t_i + t^{-1}_i$ is central; (iii) $t_0t_1t_2t_3=q^{-1}$. We display an injection of $F$-algebras $\psi :\Delta_q \to \hat H_q$ that sends $A \mapsto t_1 t_0 + (t_1 t_0)^{-1}$, $B \mapsto t_3 t_0 + (t_3 t_0)^{-1}$, $C \mapsto t_2 t_0 + (t_2 t_0)^{-1}$. For the map $\psi$ we compute the image of the three central elements mentioned above. The algebra $\Delta_q$ has another central element of interest, called the Casimir element $\Omega$. We compute the image of $\Omega$ under $\psi$. We describe how the Artin braid group $B_3$ acts on $\Delta_q$ and $\hat H_q$ as a group of automorphisms. We show that $\psi$ commutes with these $B_3$ actions. Some related results are obtained.
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